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#' Collinearity diagnostics
#'
#' @description
#' Variance inflation factor, tolerance, eigenvalues and condition indices.
#'
#' @param model An object of class \code{lm}.
#'
#' @details
#' Collinearity implies two variables are near perfect linear combinations of
#' one another. Multicollinearity involves more than two variables. In the
#' presence of multicollinearity, regression estimates are unstable and have
#' high standard errors.
#'
#' \emph{Tolerance}
#'
#' Percent of variance in the predictor that cannot be accounted for by other predictors.
#'
#' Steps to calculate tolerance:
#'
#' \itemize{
#' \item Regress the kth predictor on rest of the predictors in the model.
#' \item Compute \eqn{R^2} - the coefficient of determination from the regression in the above step.
#' \item \eqn{Tolerance = 1 - R^2}
#' }
#'
#' \emph{Variance Inflation Factor}
#'
#' Variance inflation factors measure the inflation in the variances of the parameter estimates due to
#' collinearities that exist among the predictors. It is a measure of how much the variance of the estimated
#' regression coefficient \eqn{\beta_k} is inflated by the existence of correlation among the predictor variables
#' in the model. A VIF of 1 means that there is no correlation among the kth predictor and the remaining predictor
#' variables, and hence the variance of \eqn{\beta_k} is not inflated at all. The general rule of thumb is that VIFs
#' exceeding 4 warrant further investigation, while VIFs exceeding 10 are signs of serious multicollinearity
#' requiring correction.
#'
#' Steps to calculate VIF:
#'
#' \itemize{
#' \item Regress the kth predictor on rest of the predictors in the model.
#' \item Compute \eqn{R^2} - the coefficient of determination from the regression in the above step.
#' \item \eqn{Tolerance = 1 / 1 - R^2 = 1 / Tolerance}
#' }
#'
#' \emph{Condition Index}
#'
#' Most multivariate statistical approaches involve decomposing a correlation matrix into linear
#' combinations of variables. The linear combinations are chosen so that the first combination has
#' the largest possible variance (subject to some restrictions), the second combination
#' has the next largest variance, subject to being uncorrelated with the first, the third has the largest
#' possible variance, subject to being uncorrelated with the first and second, and so forth. The variance
#' of each of these linear combinations is called an eigenvalue. Collinearity is spotted by finding 2 or
#' more variables that have large proportions of variance (.50 or more) that correspond to large condition
#' indices. A rule of thumb is to label as large those condition indices in the range of 30 or larger.
#'
#'
#' @return \code{ols_coll_diag} returns an object of class \code{"ols_coll_diag"}.
#' An object of class \code{"ols_coll_diag"} is a list containing the
#' following components:
#'
#' \item{vif_t}{tolerance and variance inflation factors}
#' \item{eig_cindex}{eigen values and condition index}
#'
#' @references
#' Belsley, D. A., Kuh, E., and Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and
#' Sources of Collinearity. New York: John Wiley & Sons.
#'
#' @examples
#' # model
#' model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
#'
#' # vif and tolerance
#' ols_vif_tol(model)
#'
#' # eigenvalues and condition indices
#' ols_eigen_cindex(model)
#'
#' # collinearity diagnostics
#' ols_coll_diag(model)
#'
#' @export
#'
ols_coll_diag <- function(model) UseMethod("ols_coll_diag")
#' @export
#'
ols_coll_diag.default <- function(model) {
check_model(model)
vift <- ols_vif_tol(model)
eig_ind <- ols_eigen_cindex(model)
result <- list(vif_t = vift, eig_cindex = eig_ind)
class(result) <- "ols_coll_diag"
return(result)
}
#' @export
#'
print.ols_coll_diag <- function(x, ...) {
cat("Tolerance and Variance Inflation Factor\n")
cat("---------------------------------------\n")
print(x$vif_t)
cat("\n\n")
cat("Eigenvalue and Condition Index\n")
cat("------------------------------\n")
print(x$eig_cindex)
}
#' @rdname ols_coll_diag
#' @export
#'
ols_vif_tol <- function(model) {
check_model(model)
vt <- viftol(model)
tibble(Variables = vt$nam,
Tolerance = vt$tol,
VIF = vt$vifs)
}
#' @rdname ols_coll_diag
#' @export
#'
ols_eigen_cindex <- function(model) {
check_model(model)
pvdata <- NULL
x <-
model %>%
model.matrix() %>%
as_data_frame()
e <-
x %>%
evalue() %>%
use_series(e)
cindex <-
e %>%
cindx()
pv <-
x %>%
evalue() %>%
use_series(pvdata) %>%
pveindex()
out <- data.frame(Eigenvalue = cbind(e, cindex, pv))
colnames(out) <- c("Eigenvalue", "Condition Index", colnames(evalue(x)$pvdata))
return(out)
}
evalue <- function(x) {
values <- NULL
y <- x
colnames(y)[1] <- "intercept"
z <- scale(y, scale = T, center = F)
tu <- t(z) %*% z
e <-
tu %>%
divide_by(diag(tu)) %>%
eigen() %>%
use_series(values)
list(e = e, pvdata = z)
}
cindx <- function(e) {
e %>%
extract(1) %>%
divide_by(e) %>%
sqrt(.)
}
#' @importFrom magrittr multiply_by_matrix
pveindex <- function(z) {
d <- NULL
v <- NULL
svdx <- svd(z)
svdxd <- svdx$d
phi_diag <-
1 %>%
divide_by(svdxd) %>%
diag()
phi <-
svdx %>%
use_series(v) %>%
multiply_by_matrix(phi_diag)
ph <-
phi %>%
raise_to_power(2) %>%
t()
diag_sum <-
ph %>%
rowSums(dims = 1) %>%
diag()
ph %>%
multiply_by_matrix(diag_sum) %>%
prop.table(margin = 2)
}
fmrsq <- function(nam, data, i) {
r.squared <- NULL
fm <-
paste0("`", nam[i], "` ", "~ .") %>%
as.formula()
m1 <-
lm(fm, data = data) %>%
summary() %>%
use_series(r.squared)
1 - m1
}
#' @description Computes vif and tolerance
#'
#' @noRd
#'
viftol <- function(model) {
m <-
model %>%
model.matrix() %>%
as_data_frame() %>%
select(-1)
nam <- names(m)
p <-
model %>%
use_series(coefficients) %>%
length() %>%
subtract(1)
tol <- c()
for (i in seq_len(p)) {
tol[i] <- fmrsq(nam, m, i)
}
vifs <- 1 / tol
list(nam = names(m), tol = tol, vifs = vifs)
}